On a $\mathbb{Z}$-module connected to approximation theory

TL;DR

This paper investigates the structure of a set related to approximation properties of real numbers, especially Pisot numbers, revealing its connection to algebraic modules and implications for number field bases.

Contribution

It describes the module structure of the set al M_ zeta and its relation to the ring of integers in algebraic number fields, extending understanding of approximation sets.

Findings

al M_ zeta lies between the ring of integers and the number field.

The module structure of al M_ zeta is characterized and compared to al O_ Q( zeta).

Results provide insights into the shape of integral bases of real number fields.

Abstract

This paper deals with the set of $\alpha\in{\mathbb{R}}$ such that $\alpha \zeta^{n} \bmod 1$ tends to $0$ for a fixed $\zeta\in{\mathbb{R}}$, which we call $\mathscr{M}_{\zeta}$. Predominately the case of Pisot numbers $\zeta$ is studied. In this case the inclusions $\mathcal{O}_{\mathbb{Q}(\zeta)}\subset\mathscr{M}_{\zeta}\subset\mathbb{Q}(\zeta)$ are known. We will show the properties of $\mathscr{M}_{\zeta}$ are connected to the module structure of the ring of integers $\mathcal{O}_{\mathbb{Q}(\zeta)}$. We will describe the module structure of $\mathscr{M}_{\zeta}$ and how much $\mathscr{M}_{\zeta}$ differs from $\mathcal{O}_{\mathbb{Q}(\zeta)}$. The results besides allow to give some information on the shape of integral bases of real number fields.